Analysis I | Part IA, 2002

State some version of the fundamental axiom of analysis. State the alternating series test and prove it from the fundamental axiom.

In each of the following cases state whether n=1an\sum_{n=1}^{\infty} a_{n} converges or diverges and prove your result. You may use any test for convergence provided you state it correctly.

(i) an=(1)n(log(n+1))1a_{n}=(-1)^{n}(\log (n+1))^{-1}.

(ii) a2n=(2n)2,a2n1=n2a_{2 n}=(2 n)^{-2}, a_{2 n-1}=-n^{-2}.

(iii) a3n2=(2n1)1,a3n1=(4n1)1,a3n=(4n)1a_{3 n-2}=-(2 n-1)^{-1}, a_{3 n-1}=(4 n-1)^{-1}, a_{3 n}=(4 n)^{-1}.

(iv) a2n+r=(1)n(2n+r)1a_{2^{n}+r}=(-1)^{n}\left(2^{n}+r\right)^{-1} for 0r2n1,n00 \leqslant r \leqslant 2^{n}-1, n \geqslant 0.

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